Cells sense substrate mechanical deformations through mechanotransduction occurring at the biomolecular scale [1]}. Such cellular signaling is carried out by proteins associated with the cell–substrate adhesions, that are in turn connected to the cell's cytoskeletal force-generating machinery [2]}. At a coarse-grained level, the contractile apparatus of cells adhered to an extracellular substrate can be modeled as active elastic inclusions [3]}, which adapts the theory of material inclusions developed by Eshelby [4]}, to describe cellular contractility as force dipoles embedded in an elastic medium. This general theoretical approach predicts how multicellular and subcellular cytoskeletal organization depend on substrate stiffness [5]}, [6]}. It has been applied successfully to explain experimental observations of substrate stiffness-dependent structural order in a variety of cell types in a unified manner [7]}, [8]}, [9]}, [10]}, [11]}. While these previous works focused on the stationary configurations of elastic dipoles in the context of adherent cells [12]}, [13]}, we now consider cell self-assembly when the cellular dipoles are free to translate and rotate in response to mechanical forces, thereby serving as minimal models for contractile cells that adhere to, spread and and crawl on soft media. We show that cell-cell mechanical interactions mediated by a compliant elastic substrate can drive network formation, and that the resulting network morphology is inherently sensitive to substrate stiffness.
